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The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

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  • Shugang Song
  • Jingjing Feng
  • Qinlong Wang

Abstract

We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold.

Suggested Citation

  • Shugang Song & Jingjing Feng & Qinlong Wang, 2014. "The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-8, July.
    • Handle: RePEc:hin:jnljam:546243
    DOI: 10.1155/2014/546243
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